Approaches to the Erd os-Ko-Rado Theorems Karen Meagher (joint - PowerPoint PPT Presentation

Approaches to the Erd˝ os-Ko-Rado Theorems Karen Meagher (joint work with Bahman Ahmadi, Peter Borg, Chris Godsil, Alison Purdy and Pablo Spiga) Finite Geometries, Irsee, September 2017

Set systems An intersecting 3-set system from { 1 , . . . , 6 } : 123 124 125 126 134 135 136 234 235 236 In this system, every set has at least 2 elements from { 1 , 2 , 3 } .

Set systems An intersecting 3-set system from { 1 , . . . , 6 } : 123 124 125 126 134 135 136 234 235 236 In this system, every set has at least 2 elements from { 1 , 2 , 3 } . Another intersecting 3-set system from { 1 , . . . , 6 } : 123 124 125 126 134 135 136 145 146 156

Set systems An intersecting 3-set system from { 1 , . . . , 6 } : 123 124 125 126 134 135 136 234 235 236 In this system, every set has at least 2 elements from { 1 , 2 , 3 } . Another intersecting 3-set system from { 1 , . . . , 6 } : 123 124 125 126 134 135 136 145 146 156 The second type of set system has many names: Trivially intersecting or Dictatorship I prefer canonically intersecting for the set of all k -sets that contain a fixed element.

Erd˝ os-Ko-Rado Theorem Theorem Let F be a t -intersecting k -set system on an n -set. If n > f ( k, t ) , then � n − t � |F| ≤ , 1 k − t

Erd˝ os-Ko-Rado Theorem Theorem Let F be a t -intersecting k -set system on an n -set. If n > f ( k, t ) , then � n − t � |F| ≤ , 1 k − t and F meets this bound if and only if it is canonically 2 t -intersecting.

Erd˝ os-Ko-Rado Theorem Theorem Let F be a t -intersecting k -set system on an n -set. If n > f ( k, t ) , then � n − t � |F| ≤ , 1 k − t and F meets this bound if and only if it is canonically 2 t -intersecting. � 3 . � k 1961 - Erd˝ os, Ko and Rado had f ( k, t ) ≥ t + ( k − t ) t 1978 - Frankl proved f ( k, t ) = ( t + 1)( k − t + 1) when t is large. 1984 - Wilson gave an algebraic proof of the bound for all t . 1997 - Ahslwede and Khachatrian detemined the largest system for all values of t , k and n .

We can ask the same question for other objects Object Definition of intersection k -Sets a common element Blocks in a design a common element Multisets a common element Vector spaces over a field a common 1-D subspace Lines in a partial geometry a common point Integer sequences same entry in same position Permutations both map i to j Permutations a common cycle Set Partitions a common class Tilings a tile in the same place Cocliques in a graph a common vertex Triangulations of a polygon a common triangle

We can ask the same question for other objects Object Definition of intersection k -Sets a common element Blocks in a design a common element Multisets a common element Vector spaces over a field a common 1-D subspace Lines in a partial geometry a common point Integer sequences same entry in same position Permutations both map i to j Permutations a common cycle Set Partitions a common class Tilings a tile in the same place Cocliques in a graph a common vertex Triangulations of a polygon a common triangle What is the size and structure of the largest set of intersecting objects?

General Framework Each object is made of k atoms . Object Atoms Sets elements from { 1 , . . . , n } Integer sequences pairs ( i, a ) (entry a is in position i ) Permutations pairs ( i, j ) (the permutation maps i to j ) Permutations cycle Set partitions subsets (cells in the partition)

General Framework Each object is made of k atoms . Object Atoms Sets elements from { 1 , . . . , n } Integer sequences pairs ( i, a ) (entry a is in position i ) Permutations pairs ( i, j ) (the permutation maps i to j ) Permutations cycle Set partitions subsets (cells in the partition) Two objects intersect if they contain a common atom. A canonically intersecting set is the set of all objects that contain a fixed atom.

General Framework Each object is made of k atoms . Object Atoms Sets elements from { 1 , . . . , n } Integer sequences pairs ( i, a ) (entry a is in position i ) Permutations pairs ( i, j ) (the permutation maps i to j ) Permutations cycle Set partitions subsets (cells in the partition) Two objects intersect if they contain a common atom. A canonically intersecting set is the set of all objects that contain a fixed atom. Objects have the EKR property if a canonically intersecting set is the largest intersecting set.

Simple Counting—Kernel Method Say we have objects and each object has k atoms.

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms.

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4 Since A is not canonical, for every i , there is an object 5 B = { b 1 , . . . , b k } in A with no a i .

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4 Since A is not canonical, for every i , there is an object 5 B = { b 1 , . . . , b k } in A with no a i . Each object in A i must contain one of the k atoms that are in B . 6

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4 Since A is not canonical, for every i , there is an object 5 B = { b 1 , . . . , b k } in A with no a i . Each object in A i must contain one of the k atoms that are in B . 6 So |A i | ≤ kP (2) , 7

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4 Since A is not canonical, for every i , there is an object 5 B = { b 1 , . . . , b k } in A with no a i . Each object in A i must contain one of the k atoms that are in B . 6 So |A i | ≤ kP (2) , and |A| ≤ k ( kP (2)) . 7

Simple Counting—Kernel Method Say we have objects and each object has k atoms. Let P (1) be the number of objects with a fixed atom; and P (2) the 1 number with 2 fixed atoms. A is a non-canonical family of intersecting objects. 2 Assume { a 1 , a 2 , . . . , a k } is an object in A . 3 A i be all the objects in A that contain the atom a i . 4 Since A is not canonical, for every i , there is an object 5 B = { b 1 , . . . , b k } in A with no a i . Each object in A i must contain one of the k atoms that are in B . 6 So |A i | ≤ kP (2) , and |A| ≤ k ( kP (2)) . 7 The objects have the EKR property, if k 2 P (2) < P (1) .

Simple Counting Bound For uniform k -partitions of { 1 , . . . , kℓ } this is k k 1 � n − iℓ � 1 � n − iℓ � k 2 � � < . ( k − 2)! ℓ ( k − 1)! ℓ i =2 i =1 Need ( k − 1) k 2 < � n − ℓ � (Works for all ℓ > 2 ). ℓ For blocks in a 2 - ( n, m, 1) design this bound is m 2 ≤ n − 1 m − 1 So any such design with m 3 − m 2 + 1 < n has the EKR property.

When Counting Fails For permutations this never works since 1 k 2 P (2) = n 2 ( n − 2)! > ( n − 1)! = P (1) . For triangulations of a convex polygon the counting never works 2 since k 2 P (2) = ( n − 3) 2 C ( n − 4) > C ( n − 3) = P (1)

When Counting Fails For permutations this never works since 1 k 2 P (2) = n 2 ( n − 2)! > ( n − 1)! = P (1) . For triangulations of a convex polygon the counting never works 2 since k 2 P (2) = ( n − 3) 2 C ( n − 4) > C ( n − 3) = P (1) In these examples the number of atoms in an object is not independent from the total number of atoms.